Asymptotics of eigenvalues and eigenfunctions of a thin square Dirichlet lattice with a curved ligament

Research output

Abstract

—The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of
the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.
Original languageEnglish
Pages (from-to)559-579
JournalMathematical Notes
Volume105
Issue number4
Early online date9 May 2019
Publication statusPublished - 2019

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